Welfare losses evaluated by response surface estimation
Taylor (1979a) argues that the tradeoff between inflation variability and output variability can be illustrated by the convex relationship in Figure 10.6. In point A monetary policy is used actively in order to keep inflation close to its target, at the expense of somewhat larger variability in output. Point C
Figure 10.6. The Taylor curve
illustrates a situation in which monetary policy responds less actively to keep the variability of inflation low, and we have smaller output variability and larger inflation variability. Point B illustrates a situation with a flexible inflation target, and we obtain a compromise between the two other points. The downward sloping curve illustrates a frontier along which the variability of output can only be brought down at the cost of increasing the variability of inflation. The preferred allocation along the Taylor curve depend on the monetary authorities’ loss function. It is, however, pointed out, for example, in Chatterjee (2002), that the Taylor curve in itself does not resolve the decision problem on which monetary policy should be adopted, and that further analysis on the welfare consequences for households of different combinations of variability of inflation and unemployment rates along the Taylor curve is required.
In the following, we will investigate how different interest rate rules behave under different choices of weights (шп, шу, ur), and under different weights A in the monetary authorities’ loss function, which we assume can be written as a linear combination of the unconditional variances of output growth A4yt and underlying inflation, A4put.
£(A) = V [Apput] + AV [Ay].
For given levels of target inflation, n*, target output growth rate g* and equilibrium real interest rate RR*, the interest rate reaction function is described by the triplet (шп, шу, шг).8 We have designed a simulation experiment in order to uncover the properties of different interest rate rules across a range of different values of these coefficients. The experiment constitutes a simple grid search across x Hy x Hr under different interest rate rules.
8 It follows that the experiment is particularly relevant for the first three types of rules in Table 10.1 (FLX, ST, and SM).
For each simulation the variance of underlying inflation, V[A4put], and output growth, V[A4yt] is calculated over the period 1995(1)-2000(4).
To summarise the different outcomes we have used the loss function £(А) — V[A4put} + W[A4yt] for А Є (0,..., 4) (11 different values).
The inflation coefficient is varied across Є (0,0.5,..., 4)(^ 9 values), the output growth coefficient is varied across шу Є (0,0.5,... , 4)(^ 9 values), and the smoothing coefficient is varied across wr Є (0, 0.1,..., 1)(^ 11 values). This makes a total of 9 x 9 x 11 = 891 simulations and 9801 loss evaluations for each type of rule/horizon.
In order to analyse such large amounts of data we need some efficient way to obtain a data reduction. We suggest to analyse the performance of the different interest rate rules by estimating a response surface for the loss function £(А) across different weights of the loss function А Є (0, 0.1,..., 1).
We consider a second-order Taylor expansion around some values, шу, wr,
and we have chosen the standard Taylor rule (0.5,0.5, 0) as our preferred choice.
£(А) ^ ao + аіШП + а^ш'у + азw'r + ві2ш'Пш'у + віз^ПШГ + візШуw'r
+ ві^'тт + ві^'у + вз w, y2 + error
Шу - — Wn, Wу — Wу Шу, Wr — Wr OJr.
a’s and в^ are estimated by OLS for each choice of weights in the loss function А Є (0, 0.1,..., 1). We minimise the estimated approximation to this loss function with respect to the three weights (шп, шу, u>r), and apply the first - order conditions to solve for these weights as functions of А in the loss function, as linear combinations of the estimated a’s and в’s.
The optimal reaction function according to this minimisation is shown in Figure 10.7, where the optimal weights, шу, wr are plotted as function of
the relative weight А assigned to output variability in the loss function. The main findings are that there is a tradeoff between variability in inflation and variability in output, irrespective of the degree of smoothing. The inflation coefficient drops as we increase the output growth weight А. Interest rate smoothing increases variability in the inflation rate without any substantial reduction in
output variability. This may explain the relatively low weight on interest rate smoothing suggested by the plot of the smoothing coefficient as a function of X.
The results from the counterfactual simulations indicate that a standard Taylor rule does quite well, across different values of the central bank preference parameters in a loss function, even in the case of a small open economy like Norway. ‘Open economy’ rules that respond to exchange rate misalignments, are shown to perform slightly worse than the Taylor rule. These rules contribute towards lower exchange rate variability without increasing interest rate variability, but at a cost of raising the variability in other target variables like headline and underlying inflation, output growth, and unemployment. Rules which respond to volatile variables like output growth produce higher interest rate volatility as a consequence. The counterfactual simulations illustrate substantial differences in the bias across the different interest rate rules, which are picked up by the RMSTE. The derivation of weights in the interest rate rules from estimated response surfaces indicate a tradeoff between variability in inflation and variability in output, irrespective of the degree of interest rate smoothing. In contrast with many other studies, interest rate smoothing seems to increase variability in the inflation rate without any substantial reduction in output variability. We conclude from this observation that statements about the optimal degree of interest rate smoothing appear to be non-robust or—to put it differently—that they are model dependent. In a situation with such conflicting evidence, the central bank should evaluate the empirical relevance and realism in the underlying models and base its decisions on the one with the highest degree of congruence.